DESY Theory Group, Hamburg

Tutorials for the Lecture

Quantum Mechanics II WS 20/21

Prof. Dr. J. Reuter Tutorial 9(Bldg. 2a/304), 3895P. Bredt/F. Fabry; B. Richard; S. van Thurenhout; T. Wening; 20.01.2021

Home Exercise 20 Relativistically covariant phase-space integration

Show for integrals for functions of 4-momentum kµ the following integralidentity:∫dkf(kµ) :=

∫d3k

(2π)3(2k0)f(kµ)

∣∣∣∣k0=+

√~k2+m2

=∫

d4k

(2π)4(2π)δ(k2−m2)θ(k0)f(kµ).

(1)In which cases is this integral Lorentz invariant?

Home Exercise 21 Kinematics: Mandelstam variables

For simple scattering reactions of relativistic elementary particles with twoparticles in initial and final state (so called 2→ 2 processes) with the kinema-tics

p1 + p2 → q1 + q2, p21 = m21, p

22 = m2

2, q21 = m2

3, q22 = m2

4 (2)it is useful to introduce the relativistically invariant Mandelstam variables:

s ≡ (p1 + p2)2 = (q1 + q2)

2 (3a)t ≡ (q1 − p1)2 = (p2 − q2)2 (3b)u ≡ (q1 − p2)2 = (p1 − q2)2 (3c)

(a) Proof the relations+ t+ u = m2

1 +m22 +m2

3 +m24. (4)

(b) We define the function

λ(a, b, c) = a2 + b2 + c2 − 2ab− 2bc− 2ca. (5)

Show that the momentum of two relativistic particles in their center-of-masssystem can be written as

~p 2CMS =

1

4sλ(s,m2

1,m22). (6)

Home Exercise 22 Vector representation of Lorentz algebra

Show that the generator for Lorentz transformation in the vector representa-tion, (Mµν)αβ = i

(gαµδνβ − gανδ

µβ

)indeed obeys the Lorentz algebra:

[Mµν ,Mρσ] = −i (gµρMνσ + gνσMµρ − gµσMνρ − gνρMµσ) .

Home Exercise 23 Creation and annihilation operators of a complex scalar field

Take the field operator of a complex scalar field in the form

φ(x) =∫dk(ake−ikx + b†ke

ikx), dk ≡ d3~k

(2π)3(2Ek), (7)

and the commutation relations of the particle and antiparticles creation andannihilation operatos

[ak, a†p] = [bk, b

†p] = (2Ek)(2π)

3δ3(~k − ~p) . (8)

Show that the creation and annihilation operators can be projected out of thefield (and adjoint field) operator in the following way:

ak = i∫d3x eikx

↔∂ t φ(x), bk = −i

∫d3x eikx

↔∂ t φ

†(x). (9)

We used the following symbol,↔∂ , which stands for

A↔∂B = A(∂B)− (∂A)B. (10)

Repetition QM I Exercise 9 Regularized incarnation of delta distribution

Show that limε→0 δε(x) ≡ limε→01

π

ε

x2 + ε2, ε > 0 is a representation of the delta

distribution/function. For this determine the function values of δε in the limitε → 0 and verify that – independent of ε – always

∫ +∞−∞ dx δε(x) = 1 holds.

The last step is to show∫ +∞−∞ dx (limε→0 δε(x))f(x) = f(0). For this, substitute

x = εξ and assume that you are allowed to exchange limit and integration.

Repetition QM I Exercise 10 Two-particle problem with central potential

Consider two particles with different masses m1 and m2, whose interactiononly depends on the relative distance of the particles (central force problem).The Hamiltonian of the system be

H =~p212m1

+~p222m2

+ V (∣∣∣~r1 − ~r2∣∣∣) . (11)

As in our setup for multi-particle quantum mechanics, operators that des-cribe observables of one of the particles commute with those of the otherparticle (”live in a different Hilbert space”),

[O1, O′2] = 0. (12)

(a) Are the orbital angular momenta of particles 1 and 2, ~L1 und ~L2 constants ofmotion? (i.e. that these operators commute componentwise with the Hamil-tonian.) If not, does a linear combination exist which is a constant of motion?

(b) Now transfer the problem to an effective one-particle problem by using aseparation ansatz. For this split the coordinates into those for the center-of-mass system and relative coordinates, and show that the total angular mo-

mentum of the system ~L = ~L1 + ~L2 can be written as

~~L = ~R× ~P + ~r × ~p, (13)

where ~R = m1~r1+m2~r2m1+m2

, ~p = m2~p1−m1~p2m1+m2

, ~r = ~r1 − ~r2 und ~P = ~p1 + ~p2. Show thatcorresponding quantities are canonically conjugated.

(c) We now add another term to Hamiltonian, a so-called spin-orbit coupling

term, which contains the total spin of the system, ~S, which fulfills the relati-ons [Si, Sj] = iεijkSk, and [Si, remainder] = 0. This adds to the Hamiltonian

for the relative motion, Hrel. a term HSL = f(|~r|)(~r× ~p) · ~S. Is ~L still a constantof motion, component-wise? If not, find a new constant of motion.